# Time-reversal-broken Weyl semimetal in the Hofstadter regime

### Submission summary

 As Contributors: Faruk Abdulla · Ankur Das · Ganpathy Murthy · Sumathi Rao Arxiv Link: https://arxiv.org/abs/2108.03196v3 (pdf) Date submitted: 2022-01-12 17:54 Submitted by: Abdulla, Faruk Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We study the phase diagram for a lattice model of a time-reversal-broken three-dimensional Weyl semimetal (WSM) in an orbital magnetic field $B$ with a flux of $p/q$ per unit cell ($0\le p \le q-1$), with minimal crystalline symmetry. We find several interesting phases: (i) WSM phases with $2q$, $4q$, $6q$, and $8q$ Weyl nodes and corresponding surface Fermi arcs, (ii) a layered Chern insulating (LCI) phase, gapped in the bulk, but with gapless surface states, (iii) a phase in which some bulk bands are gapless with Weyl nodes, coexisting with others that are gapped but topologically nontrivial, adiabatically connected to an LCI phase, (iv) a new gapped trivially insulating phase (I$'$) with (non-topological) counter-propagating surface states, which could be gapped out in the absence of crystal symmetries. Importantly, we are able to obtain the phase boundaries analytically for all $p,q$. Analyzing the gaps for $p=1$ and very large $q$ enables us to smoothly take the zero-field limit, even though the phase diagrams look ostensibly very different for $q=1, B=0$, and $q\to\infty, B\to 0$.

###### Current status:
Editor-in-charge assigned

We thank all the referees, especially referee 3, for their detailed and contructive comments which have helped us to vastly improve the manuscript. We have extensively revised the introduction and the summary, by making more explicit the novelty and context of our work, emphasized the new phases found, and discussed their implications in detail.

### List of changes

1. We have completely revised the abstract, making sure that new results are given due prominence.

2. We have thoroughly reworked the introduction in accordance with the suggestion of Referee 3. Now we provide a detailed explanation of the precise way in which our work extends previous results (minimal crystalline symmetry, field perpendicular to the line separating the Weyl nodes), and the importance of the semiclassical limit. We also list the phases we see, including the W2’ and I’ phases which are only seen at nonzero field.

3. Despite the minimal crystalline symmetry, there are a number of symmetries enjoyed by the zero-field Hamiltonian. For completeness, we have devoted an appendix to listing all these symmetries.

4. In accordance with Referee 3’s suggestion that we show the reader how the surface states decay into the bulk for the Fermi arc states and the surface states corresponding to the LCI, we have added four panels in Fig. 3, and an entirely new Fig. 4.

5. In subsection 3.3.3, we have added a paragraph to describe a topological response of W2′ phase, which is additive between the WSM surface states and the LCI surface states.

6. We have revised 3.3.4 describing the I′ phase. We offer a natural way to think about it as arising from the W2’ phase, to which it is always adjacent in the phase diagram.

7. The panels in Fig. 6 have been reorganised for clarity of presentation.

8. Panel 9c has been updated, and now shows the gap rather than its logarithm.

9. Two new references (33 and 34) concerning the annihilation of Weyl nodes in a magnetic field have been added in section 5.

10. In accordance with Referee 3’s suggestion, in the summary, the paragraphs about disorder and interactions have been completely revised, and references added.

11. A short paragraph on potential ways of experimentally realizing the model we use has been added in the summary.